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Norm on a vector space of matrices
referred to as matrix norms. Matrix norms behave in certain ways like the distance from the zero matrix. They are distinguished from the norms in general
Matrix_norm
Measure of the "size" of linear operators
{R} ^{m},} then the matrix norm given to a matrix A {\displaystyle A} is the square root of the largest eigenvalue of the matrix A ∗ A {\displaystyle
Operator_norm
Mathematical function often applied to matrices
subject to specific boundary conditions. The logarithmic norm has a wide range of applications in matrix theory, stability theory for initial and boundary value
Logarithmic_norm
Length in a vector space
identity matrix, this norm corresponds to the Euclidean norm. If A {\displaystyle A} is diagonal, this norm is also called a weighted norm. The energy norm is
Norm_(mathematics)
Largest absolute value of an operator's eigenvalues
infimum of all norms of a matrix. Indeed, on the one hand, ρ ( A ) ⩽ ‖ A ‖ {\displaystyle \rho (A)\leqslant \|A\|} for every natural matrix norm ‖ ⋅ ‖ {\displaystyle
Spectral_radius
Matrix decomposition
largest singular values of a matrix M {\displaystyle \mathbf {M} } is a matrix norm, the Ky Fan k {\displaystyle k} -norm of M {\displaystyle \mathbf
Singular_value_decomposition
Function's sensitivity to argument change
choice of norm, as can be illustrated by two examples. If ‖ ⋅ ‖ {\displaystyle \|\cdot \|} is the matrix norm induced by the (vector) Euclidean norm (sometimes
Condition_number
Real square matrix whose columns and rows are orthogonal unit vectors
In linear algebra, an orthogonal matrix or orthonormal matrix Q, is a real-valued square matrix whose columns and rows are orthonormal vectors. One way
Orthogonal_matrix
Most widely known generalized inverse of a matrix
\left(A_{n}\right)} converges to the matrix A {\displaystyle A} (in the maximum norm or Frobenius norm, say), then ( A n ) + {\displaystyle (A_{n})^{+}}
Moore–Penrose_inverse
Function that maps matrices to matrices
then the corresponding matrix series converges for matrices A such that ‖ A ‖ < r {\displaystyle \|A\|<r} for some matrix norm that satisfies ‖ A B ‖
Analytic_function_of_a_matrix
the vector norm enforcing a regularization penalty on x {\displaystyle x} has been extended to a matrix norm on X {\displaystyle X} . Matrix regularization
Matrix_regularization
Special kind of square matrix
In mathematics, a triangular matrix is a special kind of square matrix. A square matrix is called lower triangular if all the entries above the main diagonal
Triangular_matrix
Mathematical norm
the same as Schatten norm. In fact, we have ‖ A ‖ matrix p-norm , 2 = ‖ A ‖ Schatten , ∞ {\displaystyle \|A\|_{{\text{matrix p-norm}},2}=\|A\|_{{\text{Schatten}}
Schatten_norm
Topics referred to by the same term
size to any vector in a vector space Matrix norm, a map that assigns a length or size to a matrix Operator norm, a map that assigns a length or size to
Norm
Matrix representing a Euclidean rotation
rotation matrix is a transformation matrix that is used to perform a rotation in Euclidean space. For example, using the convention below, the matrix R = [
Rotation_matrix
Matrices similar to diagonal matrices
\|_{n}} is the n-Schatten norm. Note that n ≥ 2 {\displaystyle n\geq 2} Defective matrix Scaling (geometry) Triangular matrix Semisimple operator Diagonalizable
Diagonalizable_matrix
greatest common divisor matrix (sometimes abbreviated as GCD matrix) is a matrix that may also be referred to as Smith's matrix. The study was initiated
GCD_matrix
Study of matrices and their algebraic properties
particular matrix. The norm of a matrix is a positive real number. For all matrices A and B in Mmn(F), and all numbers α in F, a matrix norm, delimited
Matrix_analysis
Typographic symbol
{\displaystyle CD} " Norm: ‖ A ‖ {\displaystyle \|A\|} ; reads "the norm (length, size, magnitude etc.) of the matrix A {\displaystyle A} ". The norm of a one-dimensional
Vertical_bar
Matrix representation of a graph
theory, the Laplacian matrix, also called the graph Laplacian, admittance matrix, Kirchhoff matrix, or discrete Laplacian, is a matrix representation of a
Laplacian_matrix
Idempotent linear transformation from a vector space to itself
{\displaystyle P} and I − P {\displaystyle I-P} are equal, and thus that the matrix norm of the oblique projections are the same. However, the condition number
Projection_(linear_algebra)
Complex matrix whose conjugate transpose equals its inverse
unitary matrices with matrix multiplication forms a group, called the unitary group U(n). Every square matrix with unit Euclidean norm is the average of two
Unitary_matrix
Theorem in numerical analysis
Lax equivalence theorem. Stability in this context means that a matrix norm of the matrix used in the iteration is at most unity, called (practical) Lax–Richtmyer
Lax_equivalence_theorem
Filling in missing entries of a matrix
example, in the low-rank matrix completion problem one may apply the regularization penalty taking the form of a nuclear norm R ( X ) = λ ‖ X ‖ ∗ {\displaystyle
Matrix_completion
Square matrix where a[i,j]=1/(i+j-1)
computation. For example, the 2-norm condition number of the matrix above is about 4.8×105. Hilbert (1894) introduced the Hilbert matrix to study the following
Hilbert_matrix
Concept in mathematics
subset of n-by-n matrix units in the set of n-by-n matrices over R. The matrix norm (induced by the same two vector norms) of a matrix unit is equal to
Matrix_unit
Array of numbers
the matrix is very small. The norm of a matrix can be used to capture the conditioning of linear algebraic problems, such as computing a matrix's inverse
Matrix_(mathematics)
Matrix that commutes with its conjugate transpose
normal operators on infinite-dimensional normed spaces and to normal elements in C*-algebras. As in the matrix case, normality means commutativity is preserved
Normal_matrix
Four-dimensional number system
a diagonal complex matrix representation of complex numbers, and setting b = d = 0 produces a real matrix representation. The norm of a quaternion (the
Quaternion
American media franchise
The Matrix is an American cyberpunk media franchise consisting of four feature films, beginning with The Matrix (1999) and continuing with three sequels
The_Matrix_(franchise)
2021 film by Lana Wachowski
"these stats run 2X over the norm for the genre as far as awareness and reach." A tie-in video game tech demo titled The Matrix Awakens: An Unreal Engine
The_Matrix_Resurrections
Matrix operation generalizing exponentiation of scalar numbers
In mathematics, the matrix exponential is a matrix function on square matrices analogous to the ordinary exponential function. It is used to solve systems
Matrix_exponential
Measure of a systems floating point architecture
size of the problem, ‖ ⋅ ‖ ∞ {\displaystyle \|\cdot \|_{\infty }} is a matrix norm (largest row sum ), O ( 1 ) {\displaystyle O(1)} corresponds to the big-O
LINPACK_benchmarks
Elements of a field, e.g. real numbers, in the context of linear algebra
matrix, tensor, or other, usually, "compound" value that is actually reduced to a single component. Thus, for example, the product of a 1 × n matrix and
Scalar_(mathematics)
Concept in field theory mathematics
In mathematics, the (field) norm is a particular mapping defined in field theory, which maps elements of a larger field into a subfield. Let K be a field
Field_norm
Numerical methods for matrix eigenvalue calculation
the matrix condition number κ(A−1, b) is given by ||A||op||A−1||op, where || ||op is the operator norm subordinate to the normal Euclidean norm on Cn
Eigenvalue_algorithm
Type of matrix representation
complex matrix A {\displaystyle A} is a factorization of the form A = U P {\displaystyle A=UP} , where U {\displaystyle U} is a unitary matrix, and P {\displaystyle
Polar_decomposition
operator systems, in addition to an induced matrix norm of an operator space, one also has an induced matrix order. For operator algebras, there is still
Operator_space
theory Crouzeix's conjecture: the matrix norm of a complex function f {\displaystyle f} applied to a complex matrix A {\displaystyle A} is at most twice
List of unsolved problems in mathematics
List_of_unsolved_problems_in_mathematics
Sum of elements on the main diagonal
real matrices such that A B is a square matrix. The Frobenius inner product and norm arise frequently in matrix calculus and statistics. The Frobenius
Trace_(linear_algebra)
Formula in Lie theory
Concretely, if working with a matrix Lie algebra and ‖ ⋅ ‖ {\displaystyle \|\cdot \|} is a given submultiplicative matrix norm, convergence is guaranteed
Baker–Campbell–Hausdorff formula
Baker–Campbell–Hausdorff_formula
Mathematical concept
vector n-norm or matrix norm. Common examples of such norms include the L1 norm (sum of absolute component values), the L2 norm (Euclidean norm, or square
Approximation_error
Matrix with shifting rows
In linear algebra, a Toeplitz matrix or diagonal-constant matrix, named after Otto Toeplitz, is a matrix in which each descending diagonal from left to
Toeplitz_matrix
Particular kind of algebraic structure
sub-multiplicative matrix norm. Take the Banach space R n {\displaystyle \mathbb {R} ^{n}} (or C n {\displaystyle \mathbb {C} ^{n}} ) with norm ‖ x ‖ = max |
Banach_algebra
Hardy space Sobolev space Tsirelson space ba space Uniform norm Matrix norm Spectral radius Normed division algebra Stone–Weierstrass theorem Banach algebra
List of functional analysis topics
List_of_functional_analysis_topics
Quadric surface that looks like a deformed sphere
ISBN 978-3-642-78242-8, MR 1261419 "Lecture 15 – Symmetric matrices, quadratic forms, matrix norm, and SVD" (PDF). Archived (PDF) from the original on 2013-06-26. Retrieved
Ellipsoid
actually does not depend on the chosen matrix norm (this is true for any norm but particularly easy to see if the norm is sub-multiplicative). The joint spectral
Joint_spectral_radius
Iterative method used to solve a linear system of equations
\left\|I-\omega A\right\|\left\|e^{(k)}\right\|,} for any vector norm and the corresponding induced matrix norm. Thus, if ‖ I − ω A ‖ < 1 {\displaystyle \|I-\omega
Modified_Richardson_iteration
Unique extension of pure states in Hilbert spaces
A_{j}} by 0. The matrix norm ‖ ⋅ ‖ {\displaystyle \|\cdot \|} is the spectral norm, i.e. the operator norm with respect to the Euclidean norm on C n {\displaystyle
Kadison–Singer_problem
Matrix equal to its conjugate-transpose
In mathematics, a Hermitian matrix (or self-adjoint matrix) is a square matrix with complex-valued entries that is equal to its own conjugate transpose
Hermitian_matrix
Algorithm for modelling sequential data
linearly scaling fast weight controller (1992) learns to compute a weight matrix for further processing depending on the input. One of its two networks has
Transformer_(deep_learning)
Square matrix in which each ascending skew-diagonal from left to right is constant
In linear algebra, a Hankel matrix (or catalecticant matrix), named after Hermann Hankel, is a rectangular matrix in which each ascending skew-diagonal
Hankel_matrix
Exponential representation for differential equations
_{0}^{T}\|A(s)\|_{2}\,ds<\pi ,} where ‖ ⋅ ‖ 2 {\displaystyle \|\cdot \|_{2}} denotes a matrix norm. This result is generic in the sense that one may construct specific
Magnus_expansion
Central object in linear algebra; mapping vectors to vectors
identity matrix and N {\displaystyle \mathbf {N} } is the three-dimensional unit vector for the vector normal of the plane. If the L2 norm of a {\displaystyle
Transformation_matrix
Method of designing block ciphers
definition of a measurement for the decorrelation, which usually relies on a matrix norm; 2) the construction of simple primitive or "decorrelation module" with
Decorrelation_theory
Mathematical ring whose elements are matrices
abstract algebra, a matrix ring is a set of matrices with entries in a ring R that form a ring under matrix addition and matrix multiplication. The set
Matrix_ring
Organizational structure
specific matrix management challenges in an environment where accountability without control, and influence without authority, become the norm: Context
Matrix_management
Stochastic process
,{\sqrt {2}}Im(A_{n-1,n}))} This is an isometry, where the matrix norm is Frobenius norm. By reversing this process, a standard Brownian motion in R
Dyson_Brownian_motion
Approach in control theory to achieve fault-tolerant control for dynamic systems
A ¯ {\displaystyle {\bar {\mathbf {A} }}} in the sense of an induced matrix norm. In perfect model following, a dynamic compensator is introduced to allow
Control_reconfiguration
Square roots of the eigenvalues of the self-adjoint operator A* A
value σ 1 ( T ) {\displaystyle \sigma _{1}(T)} is equal to the operator norm of T {\displaystyle T} (see Min-max theorem). If T {\displaystyle T} acts
Singular_value
Regularization technique for ill-posed problems
{\displaystyle \mathbf {X} } is the design matrix, I {\displaystyle \mathbf {I} } is the identity matrix, and the ridge (or Tikhonov) regularization
Ridge_regression
Mathematical function
Matrix norm – Norm on a vector space of matrices Minkowski functional – Function made from a set Norm (mathematics) – Length in a vector space Normed
Seminorm
Technique in numerical linear algebra
Sylvester structured. Matrix product states, in which case the approximation is usually rescaled to have fixed Frobenius norm. The unstructured problem
Low-rank_approximation
Matrix of inner products of vectors
Gram matrix (or Gramian matrix, Gramian) of vectors v 1 , … , v n {\displaystyle v_{1},\dots ,v_{n}} in an inner product space is the Hermitian matrix of
Gram_matrix
Computer vision algorithm
algorithm used in computer vision to estimate the essential matrix or the fundamental matrix related to a stereo camera pair from a set of corresponding
Eight-point_algorithm
Algorithms for matrix decomposition
Non-negative matrix factorization (NMF or NNMF), also non-negative matrix approximation is a group of algorithms in multivariate analysis and linear algebra
Non-negative matrix factorization
Non-negative_matrix_factorization
J {\displaystyle J} . On the other hand, if the maximum matrix norm using the supremum norm for vectors of all matrices in I − F ′ ( J ) {\displaystyle
Numerical_certification
Fan norms: The sum of the k largest singular values of M is a matrix norm, the Ky Fan k-norm of M. The first of the Ky Fan norms, the Ky Fan 1-norm is
List_of_Chinese_discoveries
Mathematical theorem
{\displaystyle (\Lambda -\mu I)^{-1}V^{-1}\delta AV.} Since all p-norms are consistent matrix norms we have |λ| ≤ ||A||p where λ is an eigenvalue of A. In this
Bauer–Fike_theorem
Linear algebra concept
that it preserves the norm either in row space, or column space. A semi-orthogonal matrix always has full rank. A square matrix is semi-orthogonal if
Semi-orthogonal_matrix
reduced norm N(α) equal to mN(Ii)/N(Ij). The Brandt matrix B(m) is the H×H matrix with entries Bij. Up to conjugation by a permutation matrix it is independent
Brandt_matrix
In mathematics, invariant of square matrices
the norm N L / F : L → F {\displaystyle N_{L/F}:L\to F} of a field extension, as well as the Pfaffian of a skew-symmetric matrix and the reduced norm of
Determinant
Dictionary learning algorithm
[better source needed] Sparse approximation Singular value decomposition Matrix norm k-means clustering Low-rank approximation Michal Aharon; Michael Elad;
K-SVD
Finite dimensional algebra over a field whose central elements are that field
field to define reduced norm and reduced trace for a CSA A. Map A to a matrix ring over a splitting field and define the reduced norm and trace to be the
Central_simple_algebra
Algorithm
induced norm ‖ x ‖ B = ( ⟨ x , x ⟩ B ) 1 2 , {\displaystyle \|x\|_{B}=\left(\langle x,x\rangle _{B}\right)^{\frac {1}{2}},} and a random matrix S {\displaystyle
Kaczmarz_method
Matrix factorisation in mathematics
Schur, is a matrix decomposition. It allows one to write an arbitrary complex square matrix as unitarily similar to an upper triangular matrix whose diagonal
Schur_decomposition
Type of group in mathematics
matrices, where the group operation is given by matrix multiplication (an orthogonal matrix is a real matrix whose inverse equals its transpose). The orthogonal
Orthogonal_group
\scriptstyle 2\,\times \,2} polyphase matrix the Euclidean operator norm can be given explicitly using the Frobenius norm ‖ ⋅ ‖ F {\displaystyle \scriptstyle
Polyphase_matrix
Matrix-valued random variable
probability theory and mathematical physics, a random matrix is a matrix-valued random variable—that is, a matrix in which some or all of its entries are sampled
Random_matrix
Numerical eigenvalue calculation
eigenvalues and eigenvectors of an n × n {\displaystyle n\times n} Hermitian matrix, where m {\displaystyle m} is often but not necessarily much smaller than
Lanczos_algorithm
Study of high-dimensional data
{\Sigma }}} is a consistent estimator of Σ {\displaystyle \Sigma } in any matrix norm. When p {\displaystyle p} grows with n {\displaystyle n} , on the other
High-dimensional_statistics
Unsolved problem in matrix analysis
Crouzeix's conjecture is an unsolved problem in matrix analysis. It was proposed by Michel Crouzeix in 2004, and it can be stated as follows: ‖ f ( A )
Crouzeix's_conjecture
Arithmetic operation
division can also be defined in terms of the Hadamard product. Because matrix multiplication is not commutative, one can also define a left division or
Division_(mathematics)
Mathematical operation on invertible matrices
mathematics, a logarithm of a matrix is another matrix such that the matrix exponential of the latter matrix equals the original matrix. It is thus a generalization
Logarithm_of_a_matrix
Matrix whose conjugate transpose is its negative (additive inverse)
square matrix with complex entries is said to be skew-Hermitian or anti-Hermitian if its conjugate transpose is the negative of the original matrix. That
Skew-Hermitian_matrix
Algebraic operation on coordinate vectors
Dot product representation of a graph Euclidean norm, the square-root of the self dot product Matrix multiplication Metric tensor Multiplication of vectors
Dot_product
A CUR matrix approximation is a set of three matrices that, when multiplied together, closely approximate a given matrix. A CUR approximation can be used
CUR_matrix_approximation
Data analysis method
an L1-norm formulation, L1-PCA places linear emphasis on the coordinates of each data point, effectively restraining outliers. Consider any matrix X = [
L1-norm principal component analysis
L1-norm_principal_component_analysis
Distinguished surfaces of dynamic trajectories
| {\displaystyle \vert \,\cdot \,\vert } referring to the Euclidean matrix norm. As seen in Fig. 3, however, trajectories grow exponentially along a
Lagrangian_coherent_structure
Concepts from linear algebra
be for a matrix with two distinct eigenvalues. Geometric multiplicities are defined in a later section. For a Hermitian matrix A, the norm squared of
Eigenvalues_and_eigenvectors
Kind of linear transformation
P=(p_{ij})_{1\leq i,j\leq n}} and for all natural numbers n. Here, the respective matrix norms are naturally induced by the structure of the space of n × n {\displaystyle
Bounded_operator
Eigenvalue algorithm
norm(y) while err > epsilon x = y / norm(y); y = (A - mu * eye(rows(A))) \ x; lambda = y' * x; mu = mu + 1 / lambda err = norm(y - lambda * x) / norm(y)
Rayleigh_quotient_iteration
Elementwise product of two matrices
a matrix of the multiplied corresponding elements. This operation can be thought as a "naive matrix multiplication" and is different from the matrix product
Hadamard_product_(matrices)
Method of data analysis
pseudo-code, for a data matrix X with zero mean, without ever computing its covariance matrix. r = a random vector of length p r = r / norm(r) do c times: s
Principal_component_analysis
Inequalities in number theory and matrix theory
{\displaystyle N} differ by a matrix R {\displaystyle R} . Assume that R {\displaystyle R} is small in the sense that its spectral norm satisfies ‖ R ‖ 2 ≤ ϵ
Weyl's_inequality
Square matrix constructed from a monic polynomial
In linear algebra, the Frobenius companion matrix of the monic polynomial p ( x ) = c 0 + c 1 x + ⋯ + c n − 1 x n − 1 + x n {\displaystyle p(x)=c_{0}+c_{1}x+\cdots
Companion_matrix
}} norm. The same definition applies to H 2 {\displaystyle {\mathcal {H}}_{2}} control design. The infinity norm of the transfer function matrix F ℓ
H-infinity methods in control theory
H-infinity_methods_in_control_theory
Matrix approximation problem in linear algebra
The orthogonal Procrustes problem is a matrix approximation problem in linear algebra. In its classical form, one is given two matrices A {\displaystyle
Orthogonal_Procrustes_problem
Social norm of reciprocating actions
In social psychology, reciprocity is a social norm of responding to an action executed by another person with a similar or equivalent action. This typically
Reciprocity (social psychology)
Reciprocity_(social_psychology)
Mathematical series
conditions on the norm, then truncating the series at some finite n {\displaystyle n} may give a practical approximation to the inverse matrix: A − 1 ≈ ∑ k
Neumann_series
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